Is a tour of a US Embassy possible?

US embassies are probably among the most difficult to access but since your question suggests you might be interested in other embassies too, note that there are many options to visit one.

For example, in London, some embassies participate in the annual “Open House” event (e.g. Czech Republic, Belgium), host occasional cultural events (North Korea!) or have a public library on their premises.

In Washington, EU embassies even host their own “open house” event and you can visit the EU delegation and many EU countries' embassies on the same day (since embassies are generally pretty boring, they also organize a lunch, activities for kids, etc. at this occasion).

Other embassies located in especially significant buildings organize regular tours for tourists, e.g. the Palazzo Farnese in Rome (embassy of France) or the Hôtel de Beauharnais in Paris (official residence of the ambassador of Germany).

And of course, having a legitimate purpose (collecting a passport, voting, etc.) will give you a glimpse into your own country's embassy/consulate (although, in my experience, you generally don't see much and they often have separate facilities in the basement or a building adjacent to the embassy to receive the public).

Aside from security (mentioned by https://travel.stackexchange.com/a/44926/2387) and time (you can only go during citizen service hours), visiting a US embassy is not difficult.

You don't need to have a reason to visit the embassy and if you think you do then you can always renew your passport, request extension pages, or request they notarize some paper. Once inside you don't have to do any of these things. Since these services cost money, you can simply tell them that you don't have the money and they'll take your word for it and not provide the service.

That being said - why would you visit the embassy. It is like a DMV for expats. Do you visit the DMV in your hometown for fun? When visiting other US states, do you tour their DMV facilities?